Sequence equicontinuity, bounded sequence complexity and rigidity

Abstract In this paper, in both topological and measure-theoretical senses, various types of sequence equicontinuity and bounded sequence complexity via Furstenberg family are introduced, and the interrelationships among which and rigidity are systematically studied, which generalize many known results in the literature. In particular, it is shown that a topological dynamical system ( X , T ) is uniformly rigid iff it is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">F</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>r</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> -equicontinuous of type 4 iff it has bounded <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">F</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>r</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> -complexity of type 4; And a measure-preserving system <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">B</mml:mi> </mml:mrow> <mml:mi>X</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:mi>μ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> is rigid iff it is µ - <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">F</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>r</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> -mean equicontinuous of type 4 iff it has bounded µ - <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">F</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>r</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> -mean complexity of type 4, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">F</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>r</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> is the Furstenberg family consisting of all sequences like <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mrow> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> </mml:mrow> </mml:math> ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> </mml:mrow> </mml:math> ). Moreover, several examples are also constructed to distinguish the new notions or illustrate the scopes of some conclusions, while concurrently addressing a few questions posed in earlier articles.